\documentclass[showpacs,preprintnumbers,amsmath,amssymb,superscriptaddress,showkeys,pre]{revtex4}


\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{textcomp}
\usepackage[letterpaper,left=2.0cm,right=1.2cm,bottom=4.5cm,top=2cm]{geometry}

\usepackage{hyperref}
\newcommand{\eg}{{\it e.g.\/}}
\newcommand{\ie}{{\it i.e.\/}}
\newcommand{\etal}{{\it et al.\/}}
\hypersetup{colorlinks=true,
        %bookmarks=true,         % show bookmarks bar?
        %pdftoolbar=true,        % show Acrobat’s toolbar?
        %pdfmenubar=true,        % show Acrobat’s menu?
        citecolor=blue        % color of links to bibliography
}

\begin{document}

\title{Matrix Product States}

\author{G. Alvarez}
\affiliation{Computer Science \& Mathematics %
Division and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, %
 \mbox{Oak Ridge, TN 37831}, USA}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
tbw
\end{abstract}

\pacs{02.70.-c,05.60.Gg}% PACS, the Physics and Astronomy
                             % Classification Scheme.
\keywords{DMRG,time evolution,time-step targetting}%Use showkeys class option if keyword
                              %display desired
\maketitle

\section{\label{sec:intro}Introduction}
\subsection{Lattice growth to the right}
%$|\psi\rangle=\sum_{\sigma_1,\sigma_2} \sum_{a_1} B^{\sigma_1}_{a_1} B^{\sigma_2}_{a_1}|\sigma_1\sigma_2\rangle$
\begin{equation}
R_{b_1}^{a_1,a'_1} =  \sum_{\sigma_2,\sigma'_2} B^{\sigma_2 *}_{a_1}
W^{\sigma_2,\sigma'_{2}}_{b_1,1} B^{\sigma'_2}_{a'_1}
\end{equation}

\begin{equation}
\sum_{\sigma'_1,a'_1,b_1} W^{\sigma_1,\sigma'_1}_{1,b_1} R_{b_1}^{a_1,a'_1} M^{\sigma'_1}_{a'_1} - \lambda M^{\sigma_1}_{a_1}=0
\end{equation}

Obtain $A_{\sigma_1,a'_1}$ with an SVD: $M^{\sigma_1}_{a_1} = \sum_{a'_1} A_{\sigma_1,a'_1} S_{a'_1} (V^\dagger)_{a'_1,a_1}$.
Compute $L^{a_1,a'_1}_{b_1} = \sum_{\sigma_1,\sigma'_1} A^{\sigma_1 *}_{0,a_1} W^{\sigma_1,\sigma'_1}_{b_1} A^{0,\sigma'_1}_{a'_1}$.
\begin{equation}
R_{b_2}^{a_2,a'_2} =  \sum_{\sigma_3,\sigma'_3} B^{\sigma_3 *}_{a_2,0}
W^{\sigma_3,\sigma'_3}_{b_2,1} B^{\sigma'_3}_{a'_2,0},
\end{equation}
and from
\begin{equation}
\sum_{\sigma'_2,a'_1,a'_2,b_1,b_2} L^{a_1,a'_1}_{b_1} W^{\sigma_2,\sigma'_2}_{b_1,b_2} R_{b_2}^{a_2,a'_2} M^{\sigma'_2}_{a'_1,a'_2}
- \lambda M^{\sigma_2}_{a_1,a_2}=0
\end{equation}
obtain $M^{\sigma_2}_{a_1,a_2}$ and via an SVD $A^{\sigma_2}_{a_1,a'_2}$ where
 \begin{equation}
M^{\sigma_2}_{a_1,a^B_2}=M_{(a1\sigma_2)a^B_2}=\sum_{a^{'A}_2} A_{(a1,\sigma_2),a^{'A}_2} S_{a^{'A}_2}(V^\dagger)_{a^{'A}_2,a_2}
\end{equation}
\begin{equation}
L^{a_2,a'_2}_{b_2} = \sum_{a_1,b_1,a'_1} L^{a_1,a'_1}_{b_1} \sum_{\sigma_2\sigma'_2} A^{\sigma_2 *}_{a_1,a_2}
W^{\sigma_2\sigma'_2}_{b_1,b_2}A^{\sigma'_2}_{a'_1,a'_2}
\end{equation}

\subsection{Sweep to the left}

Recompute $B$ according to:

 \begin{equation}
M^{\sigma_2}_{a_1,a^B_2}=M_{(\sigma_2a^B_2),a_1}=\sum_{a^{'B}_2} B_{(\sigma_2a^B_2),a^{'B}_2} S_{a^{'B}_2}(V^\dagger)_{a^{'B}_2,a_1}
\end{equation}

Compute $R$:
\begin{equation}
R^{a_{L-2},a'_{L-2}}_{b_{L-2}} = \sum_{a_{L-1},b_{L-1},a'_{L-1}}\sum_{\sigma_{L-1},\sigma'_{L-1}} B^{\sigma_{L-1}*}_{a_{L-2},a_{L-1}}
W^{\sigma_{L-1},\sigma'_{L-1}}_{b_{L-2},b_{L-1}} B^{\sigma'_{L-1}}_{a'_{L-2},a'_{L-1}} R^{a_{L-1},a'_{L-1}}_{b_{L-1}}
\end{equation}

\end{document}
%
% ****** End of file apssamp.tex ******
